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Simplifying 4k2 + -104k = -84 Reorder the terms: -104k + 4k2 = -84 Solving -104k + 4k2 = -84 Solving for variable 'k'. Reorder the terms: 84 + -104k + 4k2 = -84 + 84 Combine like terms: -84 + 84 = 0 84 + -104k + 4k2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(21 + -26k + k2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(21 + -26k + k2)' equal to zero and attempt to solve: Simplifying 21 + -26k + k2 = 0 Solving 21 + -26k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '-21' to each side of the equation. 21 + -26k + -21 + k2 = 0 + -21 Reorder the terms: 21 + -21 + -26k + k2 = 0 + -21 Combine like terms: 21 + -21 = 0 0 + -26k + k2 = 0 + -21 -26k + k2 = 0 + -21 Combine like terms: 0 + -21 = -21 -26k + k2 = -21 The k term is -26k. Take half its coefficient (-13). Square it (169) and add it to both sides. Add '169' to each side of the equation. -26k + 169 + k2 = -21 + 169 Reorder the terms: 169 + -26k + k2 = -21 + 169 Combine like terms: -21 + 169 = 148 169 + -26k + k2 = 148 Factor a perfect square on the left side: (k + -13)(k + -13) = 148 Calculate the square root of the right side: 12.165525061 Break this problem into two subproblems by setting (k + -13) equal to 12.165525061 and -12.165525061.Subproblem 1
k + -13 = 12.165525061 Simplifying k + -13 = 12.165525061 Reorder the terms: -13 + k = 12.165525061 Solving -13 + k = 12.165525061 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + k = 12.165525061 + 13 Combine like terms: -13 + 13 = 0 0 + k = 12.165525061 + 13 k = 12.165525061 + 13 Combine like terms: 12.165525061 + 13 = 25.165525061 k = 25.165525061 Simplifying k = 25.165525061Subproblem 2
k + -13 = -12.165525061 Simplifying k + -13 = -12.165525061 Reorder the terms: -13 + k = -12.165525061 Solving -13 + k = -12.165525061 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '13' to each side of the equation. -13 + 13 + k = -12.165525061 + 13 Combine like terms: -13 + 13 = 0 0 + k = -12.165525061 + 13 k = -12.165525061 + 13 Combine like terms: -12.165525061 + 13 = 0.834474939 k = 0.834474939 Simplifying k = 0.834474939Solution
The solution to the problem is based on the solutions from the subproblems. k = {25.165525061, 0.834474939}Solution
k = {25.165525061, 0.834474939}
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